Understanding a proof of Klein Bottle being embeddable in $\mathbb R^4$ Klein Bottle being embeddable in $\mathbb R^4$:
Here it is a proof in the book 'C Adams and R Franzosa - Introduction to Topology. Pure and Applied': 

My question is how introducing $h(x)$ solves the problem; I mean:
1- By introducing $h(x)$, how the two $D$'s in the figure could be avoided to be same if we go to $\mathbb R^4$? (As intuitively, we can't cross a river unless we hump it, i.e. going to the extra dimension - height)
2- How to prove that the function $F:K \rightarrow \mathbb R^4$ is continuous and bijective?
PS I couldn't find the above-mentioned proof in other questions of MSE. 
 A: The function $h$ is the "hump over the river" that you need to embed $K$ in $\mathbb R^4$. It provides the extra information necessary to distinguish the points that the failed embedding $f$ identifies. For it to do that though, one needs to guarantee (as stated) that the points identified by $f$ have different values under $h$.
The reason for this is the $\mathbb R^4$ embedding $F$, which is defined by $F(x) = (f(x),h(x))$.
If $x_1$ and $x_2$ are identified by $f$, then the first (three) coordinate(s) of $F(x_1)$ and $F(x_2)$ will agree, so we need to guarantee that $h(x_1)$ and $h(x_2)$ are distinct.
You will not be able to show that $F$ is a bijection, because it is not.
What I assume you meant was to show that $F$ is a homeomorphism of $K$ and the image $F(K)$.
Showing that $F$ is continuous is straightforward: you already know that $f$ is continuous so once you have know that $h$ is continuous, you can use that a function such as $F$ is continuous if and only if its coordinate functions are.
All that remains is for you to do is to show that $F$ is injective (since of course it is onto its image) and that the inverse $F^{-1}$ is also continuous.
